Phase modulation

The modulating wave (blue) is modulating the carrier wave (red), resulting the PM signal (green). g(t) = π/2 * sin(2*2πt+ π/2*sin(3*2πt))

PM changes the phase angle of the complex envelope in direct proportion to the message signal.

If m(t) is the message signal to be transmitted and the carrier onto which the signal is modulated is

${\displaystyle c(t)=A_{c}\sin \left(\omega _{\mathrm {c} }t+\phi _{\mathrm {c} }\right).}$,

then the modulated signal is

${\displaystyle y(t)=A_{c}\sin \left(\omega _{\mathrm {c} }t+m(t)+\phi _{\mathrm {c} }\right).}$

This shows how ${\displaystyle m(t)}$ modulates the phase - the greater m(t) is at a point in time, the greater the phase shift of the modulated signal at that point. It can also be viewed as a change of the frequency of the carrier signal, and phase modulation can thus be considered a special case of FM in which the carrier frequency modulation is given by the time derivative of the phase modulation.

The modulation signal could here be

${\displaystyle m(t)=\cos \left(\omega _{\mathrm {c} }t+h\omega _{\mathrm {m} }(t)\right)\ }$

The mathematics of the spectral behavior reveals that there are two regions of particular interest:

• For small amplitude signals, PM is similar to amplitude modulation (AM) and exhibits its unfortunate doubling of baseband bandwidth and poor efficiency.
• For a single large sinusoidal signal, PM is similar to FM, and its bandwidth is approximately

${\displaystyle 2\left(h+1\right)f_{\mathrm {M} }}$,

where ${\displaystyle f_{\mathrm {M} }=\omega _{\mathrm {m} }/2\pi }$ and ${\displaystyle h}$ is the modulation index defined below. This is also known as Carson's Rule for PM.

As with other modulation indices, this quantity indicates by how much the modulated variable varies around its unmodulated level. It relates to the variations in the phase of the carrier signal:

${\displaystyle h\,=\Delta \theta \,}$,

where ${\displaystyle \Delta \theta }$ is the peak phase deviation. Compare to the modulation index for frequency modulation.

• Angle modulation
• Automatic frequency control
• Modulation for a list of other modulation techniques
• Modulation sphere
• Polar modulation
• Electro-optic modulator for Pockel's Effect phase modulation for applying sidebands to a monochromatic wave